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Transcript
MA: Binomial Expansion (1st technique: Pascal’s Triangle) 1 1 1 1 1 __ 2 3 4 __ 1 1 3 6 __ 1 4 __ 1 __ __ These numbers are the coefficients for each term in a binomial expansion. (a b)0 1a0b0 (a b)1 1a1b0 1a0b1 (a b)2 1a 2b0 2a1b1 1a 0b2 (a b)3 1a3b0 3a 2b1 3a1b2 1a0b3 (a b)4 1a 4b0 4a3b1 6a 2b2 4a1b3 1a 0b4 What will the next expansion be? (a b)5 Use the pattern in Pascal’s triangle to write ( x y)7 in expanded form. Binomial Expansion (2nd technique: Binomial Theorem) Before beginning we need to know the notation C (known as a Combination) n r n Cr n! n r !r ! Example: Find the value of C : 11 5 3 C5 : The Binomial Theorem: n In the expansion of (x + y) n n n–1 (x + y) = x +nx n–r r n–1 y + … + C x y + … + nxy n r n +y n–r r The coefficient of x y is n Cr n! n r !r ! n The symbol is often used in place of C to denote n r r binomial coefficients. Ex 2: Expand ( x y )3 using the Binomial Theorem: ( x y )3 = 3 C0 x3 Ex 3: Expand (2 x y)5 using the Binomial Theorem: Finding a certain term of a binomial expansion: Example: Find the 4th term of ( x 2 y)6 : Here n = 6. To find r, use: (term number desired) – 1. Therefore r = 4-1 = 3. 6 C3 x 63 (2 y )3 1) Find the 3rd term of (2 x y)6 : 2) Find the 5th term of ( x 1)9 :